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\author{Class 2019 Math and Applied Math }
\title{Applied stochastic processes - Homework 03}
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%\date{2021 年 2 月 28 日}
\date{March 23, 2021}
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%\subsection{Homework 03}
%E4.1.10, P4.1.1, P4.1.5, E4.2.6, E4.3.2, E4.4.2.

\begin{document}

\maketitle

\begin{enumerate}

\item [E4.1.10.] A bus in a mass transit system is operating on a continuous route with intermediate stops. The arrival of the bus at a stop is classified into one of three states, namely: 1. Early arrival; 2. On-time arrival; 3. Late arrival. 
Suppose that the successive states form a Markov chain with transition probability matrix below. Over a long period of time, what fraction of stops can be expected to be late?
\begin{eqnarray*}
P=
\begin{blockarray}{cccc}
& 1 & 2 & 3 \\
\begin{block}{c[ccc]}
  1   & 0.5 & 0.4 & 0.1 \\ 
  2   & 0.2 & 0.5 & 0.3 \\
  3   & 0.1 & 0.2 &0.7 \\
\end{block}
\end{blockarray}.
\end{eqnarray*}


\item [P4.1.1.] Five balls are distributed between two urns, labeled A and B. Each period, an urn is selected at random, and if it is not empty, a ball from that urn is removed and placed into the other urn. In the long run what fraction of time is urn A empty?


\item [P4.1.5.] The four towns A,B,C, and D are connected by railroad lines as shown in the following diagram. Each day, in whichever town it is in, a train chooses one of the lines out of that town at random and traverses it to the next town, where the process repeats the next day. In the long run, what is the probability of finding the train in town D?

\begin{center}
\tikz{
\node [circle, draw] (c) at (4,-1) {C}; 
\node [circle, draw] (d) at (2,0) {D}; 
\node [circle, draw] (b) at (2,-1) {B}; 
\node [circle, draw] (a) at (0,-1) {A}; 

\graph {(a) --  (b) };
\graph {(a) --  (d) };
\graph {(b) --  (d) };
\graph {(b) --  (c) };
}
\end{center}

\item [E.4.2.6.] A component of a computer has an active life, measured in discrete units, that
is a random variable $T$ where
$
\mathbb{P}\{T =1\}=0.1, \,\,\,
\mathbb{P}\{T =3\}=0.3, \,\,\,
\mathbb{P}\{T =2\}=0.2, \,\,\,
\mathbb{P}\{T =4\}=0.4. 
$
Suppose one starts with a fresh component, and each component is replaced by a new component upon failure. Determine the long run probability that a failure occurs in a given period.


\item [E.4.3.2.] Which states are transient and which are recurrent in the Markov chain whose transition probability matrix is $P$ below. 
\begin{eqnarray*}
P=
\begin{blockarray}{ccccccc}
        & 0 &   1 &   2 &     3 &  4 &    5 \\
\begin{block}{c[cccccc]}
  0   & 1/3 &   0  & 1/3  &  0  &  0  &  1/3  \\ 
  1   & 1/2 & 1/4 & 1/4  &  0  &  0  &  0  \\ 
  2   & 0    &   0  & 0     &  0  &  1  &  0  \\ 
  3   & 1/4 & 1/4  & 1/4  &  0  &  0  & 1/4  \\ 
  4   &  0  &   0   &   1  &  0  &  0  &  0  \\ 
  5   &  0  &   0  &    0  &  0  &  0  &  1  \\ 
\end{block}
\end{blockarray}.
\hspace{1cm}
Q=
\begin{blockarray}{ccccc}
 & 0 & 1 & 2 & 3 \\
\begin{block}{c[cccc]}
  0   & 1    & 0    & 0  & 0 \\ 
  1   & 0.1 & 0.4 & 0.2 & 0.3 \\ 
  2   & 0.2 & 0.2 & 0.5 & 0.1 \\
  3   & 0.3 & 0.3 &0.4 & 0 \\
\end{block}
\end{blockarray}.
\end{eqnarray*}


\item [E4.4.2.] Consider the Markov chain whose transition probability matrix is given by $Q$ above. 
\begin{enumerate}
\item  Determine the limiting probability $\pi_0$ that the process is in state 0.
\item  By pretending that state 0 is absorbing, use a first step analysis and calculate the mean time $m_{10}$ for the process to go from state 1 to state 0.
\item  Because the process always goes directly to state 1 from state 0, the mean return time to state 0 is $m_0 = 1 + m_{10}$. Verify the equation $\pi_0 = 1/m_0$.
\end{enumerate}


\end{enumerate}


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\end{document}

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\subsection{Homework 01}
E3.1.2, P3.1.4, E3.2.2, P3.2.4, E3.3.2, P3.3.6.

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\subsection{Homework 02}
E.3.4.1, E3.4.2, P3.4.1, P3.4.5, E3.5.1, P3.5.1. 

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\subsection{Homework 03}
E4.1.10, P4.1.1, P4.1.5, E4.3.1, E4.3.2, E4.4.2.

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\subsection{Homework 04}
E5.1.1, E5.1.7, P5.1.10, E5.2.1, P5.2.1.

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\subsection{Homework 05}
E5.3.1, E5.3.3, E5.3.7, P5.3.1, E5.4.1, E5.4.3. 

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\subsection{Homework 06}
E6.1.1, E6.1.2, P6.1.1, P6.1.2.

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\subsection{Homework 07}
E7.1.2, E7.1.3, E7.2.1, E7.2.3, P7.2.1.

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\subsection{Homework 08}
E8.1.1, E8.1.2, E8.1.4, P8.1.1, P8.1.3, E8.2.1.

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